Pluralities and Sets

Abstract

Say that some things form a set just in case there is a set whose members are precisely the things in question. For instance, all the inhabitants of New York form a set. So do all the stars in the universe. And so do all the natural numbers. Under what conditions do some things form a set? Let us make the question more precise. Plural expressions such as ‘some things’ are best explained by means of plural logic. In ordinary singular first-order logic we have singular variables such as x and y, which can be bound by the existential and universal quantifiers. In plural first-order logic we have in addition plural variables such as xx and yy, which can also be bound by existential and universal quantifiers to yield so-called plural quantifiers.1 ‘∃xx’ is read as ‘there are some things xx such that . . . ’, and ‘∀xx’, as ‘given any things xx, . . . ’. These quantifiers are subject to inference rules analogous to those of the ordinary singular quantifiers. There is also a two-place logical predicate ≺, where ‘u ≺ xx’ is to be read as ‘u is one of xx’. By ‘set’ I mean set as on the standard iterative conception, according to which the sets are “formed” in stages.2 We begin at stage 0 with all the non-sets, which are known as ∗I am grateful to a large number of people for comments and discussion, in particular Daniel Isaacson, Jose Martinez, David Nicolas, Vann McGee, Richard Pettigrew, Agustin Rayo, Gabriel Uzquiano, Tim Williamson, and two anonymous referees, as well as audiences in Bristol, Buenos Aires, MIT, Munich, Oxford, Riga, and Southampton. (At some of these places an earlier version was presented under the title “Why size does not matter.”) Much of the paper was written during a period of research leave funded by the AHRC (grant AH/E003753/1), whose support I gratefully acknowledge. Plural first-order logic was made popular by George Boolos, “To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables),” this Journal, LXXXI (1984): 430-439. For an introduction, see Oystein Linnebo, “Plural Quantification,” in Edward N. Zalta, ed., Stanford Encyclopedia of Philosophy (2008). See George Boolos, “The Iterative Conception of Set,” this Journal, LXVIII (1971):215-32. It is of course controversial how the metaphor of “set formation” should be understood. I will provide a modal explication in Section 5, following Charles Parsons, “What Is the Iterative Conception of Set?,” repr. in his Mathematics in Philosophy (Ithaca: Cornell, 1983), pp. 268-297.